Question

Find the general term and the sum of n terms of the series $-3,-1,11,39,89,167,....$

Answer 1

$\Rightarrow -3-1113989\mathrm{167....}$

$2122850\mathrm{78....}$

$101622\mathrm{28....}$

$66\mathrm{6....}$

Assuming $u_n=A+Bn+C{n}^2+D{n}^3$

Putting $n=1,2,3,4$, we get;-

$u_n=n^3-{(n+1)}^2$

$=n^3-n-(n^2+n+1)$

$=(n-1)n(n+1)-n(n+1)-1$

$S_n=\sum u_n$

$S_n=\frac{1}{4}(n-1)\left(n\right)(n+1)(n+2)-\frac{1}{3}n(n+1)(n+2)-n$